Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{y^2 - 8y}{y^2 - 16y + 64}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - 8y}{y^2 - 16y + 64} = \dfrac{(y)(y - 8)}{(y - 8)(y - 8)} $ Notice that the term $(y - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y - 8)$ gives: $r = \dfrac{y}{y - 8}$ Since we divided by $(y - 8)$, $y \neq 8$. $r = \dfrac{y}{y - 8}; \space y \neq 8$